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Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

Michał Lipiński, Jacek Kubica, Marian Mrozek, Thomas Wanner

Abstract

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a unique maximal element. The extension is from the setting of Lefschetz complexes to the more general situation of finite topological spaces. We define isolated invariant sets, isolating neighbourhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.

Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

Abstract

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a unique maximal element. The extension is from the setting of Lefschetz complexes to the more general situation of finite topological spaces. We define isolated invariant sets, isolating neighbourhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.

Paper Structure

This paper contains 27 sections, 76 theorems, 110 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A simple combinatorial flow on a simplicial complex
  4. Preliminaries
  5. Sets and maps
  6. Relations and digraphs
  7. Posets
  8. Finite topological spaces
  9. Homology of finite topological spaces
  10. Dynamics of combinatorial multivector fields
  11. Multivalued dynamical systems in finite topological spaces
  12. Solutions and paths
  13. Combinatorial multivector fields
  14. Essential solutions
  15. Isolated invariant sets
  16. ...and 12 more sections

Key Result

Proposition 3.1

Let $(X,\leq)$ be a poset and let $A, B\subset X$.Then Moreover, if $A$ and $B$ are down sets, then

Figures (11)

  • Figure 1: An example of a simplicial complex (top) and the poset (a finite $T_0$ topological space, bottom) induced by its face relation.
  • Figure 2: A partition of a poset into multivectors (convex subsets). Nodes as well as corresponding arrows of each multivector are highlighted with a distinct color.
  • Figure 3: A geometric visualization of the combinatorial multivector field in Figure \ref{['fig:mvf-poset-example']}. A multivector may be considered as a "black box" whose dynamics is known only via splitting its boundary into the exit and entrance parts.
  • Figure 4: The combinatorial flow $\Pi_\text{$\mathcal{V}$}$ of the multivector field in Figures \ref{['fig:mvf-poset-example']} and \ref{['fig:mvf-complex-example']} represented as the digraph $G_\text{$\mathcal{V}$}$. Downward arrows are induced by the closure components of $\Pi_\text{$\mathcal{V}$}$. Bi-directional edges and self-loops reflect dynamics within multivectors. For clarity, we omit edges that can be obtained by between-level transitivity, e.g., the bi-directional connection between node $D$ and $BCD$. The nodes of critical multivectors are bolded in red.
  • Figure 5: The Conley-Morse graph for the example in Figure \ref{['fig:mvf-graph-example']}.
  • ...and 6 more figures

Theorems & Definitions (132)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 3.8
  • ...and 122 more